CRB: Sinusoid-Sources' Estimation using Collocated Dipoles/Loops

This work derives new asymptotic Cramer-Rao lower bounds (CRB) for the estimation of multiple pure-tone incident signals' azimuth-elevation arrival-angles, polarization parameters, frequencies, amplitudes, and temporal phases-based on data collected by spatially collocated but orthogonally oriented dipoles and/or loops. The incident sources are pure-tones at distinct, deterministic but unknown frequencies, in contrast to the case of all incident sources at one common known frequency, as has been investigated in the existing research literature on the CRB for diversely-polarized direction-finding. The derived CRBs are closed-form expressions, explicitly in terms of the signal parameters. The new CRBs presented here reveal how a constituent dipole and/or loop's presence and orientation may affect estimation precision, thereby offering guidelines to the system engineer on what dipole(s) and/or loop(s) to include or to omit in constructing the electromagnetic vector-sensor.

I. INTRODUCTION11This Paper will Use the Following Notation: $(\cdot)^{\rm T}$ for Transposition, $(\cdot)^{\rm H}$ for the Hermitian Operation, Diag $(x_{1},\ldots, x_{N})$ for an $N\times N$. Diagonal Matrix with $x_{1},\ldots, x_{N}$ as the Diagonal Elements, Re ${\rm Re}(\cdot)$ for a Complex entity's Real-Value Part, $\otimes$ for the Kronecker Product, ° for an Element-Wise (Hadamard) Product Operator, for $\sqrt{-1}, {\bf I}_{N\times N}$ for an $N\times N$. Identity Matrix, ${\rm vec}(\cdot)$ for Stacking a matrix's Columns into One Column Vector, and $O(N^{i})$ for an Order-Of-Magnitude Same as $N^{i}$. Where $i$ is an Integer.

This Paper will Use the Following Notation: for Transposition, for the Hermitian Operation, Diag for an . Diagonal Matrix with as the Diagonal Elements, Re for a Complex entity's Real-Value Part, for the Kronecker Product, ° for an Element-Wise (Hadamard) Product Operator, for for an . Identity Matrix, for Stacking a matrix's Columns into One Column Vector, and for an Order-Of-Magnitude Same as . Where is an Integer.